We propose a dimensional regularization scheme to deal with the divergences caused by colliding supercurrents inserted at the interaction points, in the light-cone gauge NSR superstring field theory. We formulate the theory in $d$ dimensions and define the amplitudes as analytic functions of $d$. With an appropriately chosen three-string interaction term and large negative $d$, the tree level amplitudes for the (NS,NS) closed strings can be recast into a BRST invariant form, using the superconformal field theory proposed in Ref.[arXiv:0911.3704]. We show that in the limit $d to 10$ they coincide with the results of the first quantized theory. Therefore we obtain the desired results without adding any contact interaction terms to the action.
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